# Snowball Problem: Differential Equation of Translational Motion

• sheelbe999
In summary, the snowball's translational motion can be described by the differential equation F=mg sin(*)=m dv/dt + a v^2, where the additional mass accumulated is proportional to the distance traveled and friction and rotation are ignored. The equation takes into account the forces acting on the snowball and its acceleration as it slides down the inclined slope.

## Homework Statement

"A snowball, initially of mass m, slides down a slope inclined at an angle φ with
respect to the horizontal. As it moves, the mass of additional snow Δm = αx that it
accumulates is proportional to the distance traveled x. Write the differential equation
of translational motion for the snowball, ignoring rotation and friction. [6]"

I was sure mass was irrelevant in this situation? But judging by the question its not. Could anyone help me out?

## The Attempt at a Solution

As always, I'd begin by drawing a picture; what Force(s) act on the snowball, and which direction do they point?

F=d(mv)/dt
F=v dm/dt + m dv/dt

m=m+ax
dm/dt=av

F=mg sin(*)

mg sin(*)=m dv/dt + a v^2

## 1. What is the Snowball Problem: Differential Equation of Translational Motion?

The Snowball Problem: Differential Equation of Translational Motion is a mathematical problem that involves finding the motion of a snowball rolling down a hill. It uses differential equations to model the motion of the snowball and determine its position, velocity, and acceleration at any given time.

## 2. Why is the Snowball Problem important in science?

The Snowball Problem is important in science because it allows us to understand the behavior of objects in motion and how they are affected by external forces. This problem can be applied to real-life situations, such as predicting the trajectory of a moving object or designing efficient transportation systems.

## 3. What are the key components of the Snowball Problem: Differential Equation of Translational Motion?

The key components of the Snowball Problem are the initial conditions, external forces acting on the snowball, and the differential equation that describes the motion of the snowball. The initial conditions include the starting position, velocity, and acceleration of the snowball. The external forces can include gravity, friction, and air resistance.

## 4. How is the Snowball Problem solved?

The Snowball Problem is typically solved using integration techniques to solve the differential equation. The initial conditions and external forces are used to set up the equation, and then numerical or analytical methods can be used to find the solution. Technology, such as computer software, can also be used to solve the problem.

## 5. What are some real-life applications of the Snowball Problem: Differential Equation of Translational Motion?

The Snowball Problem has many real-life applications in fields such as physics, engineering, and transportation. It can be used to design roller coasters, predict the trajectory of projectiles, and optimize the motion of vehicles. Understanding the principles behind this problem can also help in the development of more efficient and accurate models for predicting the motion of objects in various scenarios.

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